In this paper, we study bi-center and bi-isochronous center problems in cubic planar systems which are symmetric with respect to a straight line. These systems can be transformed to ones which are symmetric with respect to the y-axis and have two symmetric singular points at (±1,0), which can be classified as elementary and nilpotent singular points. A complete classification is given on the centers, including nine conditions for elementary singular points and four conditions for nilpotent singular points. Moreover, seven bi-isochronous center conditions are obtained for the elementary singular points.